74 1. ALGEBRAIC THEMES group of rotations of the square card. Show that the bijection e $ 1 r $ i r 2 $ ± 1 r 3 $ ± i produces a matching of the multiplication tables of the two groups. That is, if we apply the bijection to each entry of the multiplication table of H , we produce the multiplication table of R . Thus, the two groups are isomorphic. 1.10.4. Show that the group C 4 D f i; ± 1; ± i;1 g of fourth roots of unity in the complex numbers is isomorphic to Z 4 . The next several exercises give examples of groups coming from var-ious areas of mathematics and require some topology or real and complex analysis. Skip the exercises for which you do not have the appropriate background. 1.10.5. An isometry of R 3 is a bijective map T W R 3 ! R 3 satisfying d.T.x/;T.y// D d.x;y/ for all x;y 2 R 3 . Show that the set of isome-tries of R 3 forms a group. (You can replace R 3 with any metric space .) 1.10.6. A homeomorphism of R 3 is a bijective map T W R 3 ! R 3 such that both T and its inverse are continuous. Show that the set of homeomor-
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